# How to transform a $U(0,1)$ variable to produce a Poisson variable?

Suppose $$X$$ is a uniformly distribution over $$(0,1)$$. How to find transformations $$Y=g(X)$$ to produce random variables with the Poisson distribution?

You can go for $$g:=F^{-1}$$ where $$F$$ denotes the CDF of this Poisson distribution.
In general if $$X$$ has uniform distribution on $$(0,1)$$ and $$F$$ is a CDF of some distribution then it can be shown that random variable $$Y:=F^{-1}(X)$$ has function $$F$$ as CDF.
Here $$F^{-1}:(0,1)\to\mathbb R$$ is prescribed by:$$u\mapsto\inf\{x\in\mathbb R\mid F(x)\geq u\}$$